Space-Efficient Fragments of Higher-Order Fixpoint Logic Florian - - PowerPoint PPT Presentation

Space-Efficient Fragments of Higher-Order Fixpoint Logic

Florian Bruse 1 2 Martin Lange 1 Etienne Lozes 2

1Universit¨

at Kassel, Germany

2LSV, ENS Paris-Saclay, France

September 8, 2017

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Motivation

Higher-order Modal Fixpoint Logic:

• extension of the modal µ-calculus by simply typed λ-calculus
• very expressive logic, e.g., context-sensitive path languages,

assume-guarantee properties

• captures time hierarchy: type order k + 1 model checking is

k + 1-EXPTIME complete (via alternating reachability game)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Motivation

Higher-order Modal Fixpoint Logic:

• extension of the modal µ-calculus by simply typed λ-calculus
• very expressive logic, e.g., context-sensitive path languages,

assume-guarantee properties

• captures time hierarchy: type order k + 1 model checking is

k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy?

2 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Motivation

Higher-order Modal Fixpoint Logic:

• extension of the modal µ-calculus by simply typed λ-calculus
• very expressive logic, e.g., context-sensitive path languages,

assume-guarantee properties

• captures time hierarchy: type order k + 1 model checking is

k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment:

• restrict interplay of fixpoints, mixing with certain operators
• model-checking for order k + 1 in k-EXPSPACE (via nondet.

reachability game)

2 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Motivation

Higher-order Modal Fixpoint Logic:

• extension of the modal µ-calculus by simply typed λ-calculus
• very expressive logic, e.g., context-sensitive path languages,

assume-guarantee properties

• captures time hierarchy: type order k + 1 model checking is

k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment:

• restrict interplay of fixpoints, mixing with certain operators
• model-checking for order k + 1 in k-EXPSPACE (via nondet.

reachability game)

• matching lower bound via reduction to order-k corridor tiling

problem

2 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Motivation

Higher-order Modal Fixpoint Logic:

• extension of the modal µ-calculus by simply typed λ-calculus
• very expressive logic, e.g., context-sensitive path languages,

assume-guarantee properties

• captures time hierarchy: type order k + 1 model checking is

k + 1-EXPTIME complete (via alternating reachability game) Question: Can we capture space hierarchy? → Tail-recursive fragment:

• restrict interplay of fixpoints, mixing with certain operators
• model-checking for order k + 1 in k-EXPSPACE (via nondet.

reachability game)

• matching lower bound via reduction to order-k corridor tiling

problem

• still very expressive: anbncn path language

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Syntax of HFL

HFL = modal µ-calculus ϕ ::= P | X | ϕ∨ϕ | ¬ϕ | aϕ | µ X .ϕ plus duals ϕ ∧ ψ, [a]ϕ, ν X natively

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Syntax of HFL

HFL = modal µ-calculus + simply typed λ-calculus

[Viswanathan2 ’04]

ϕ ::= P | X | x | ϕ∨ϕ | ¬ϕ | aϕ | µ X .ϕ | λ x .ϕ | ϕ ϕ plus duals ϕ ∧ ψ, [a]ϕ, ν X natively

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Syntax of HFL

HFL = modal µ-calculus + simply typed λ-calculus

[Viswanathan2 ’04]

ϕ ::= P | X | x | ϕ∨ϕ | ¬ϕ | aϕ | µ X .ϕ | λ x .ϕ | ϕ ϕ plus duals ϕ ∧ ψ, [a]ϕ, ν X natively But expressions like aP bP not well defined

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Syntax of HFL

HFL = modal µ-calculus + simply typed λ-calculus

[Viswanathan2 ’04]

ϕ ::= P | X | x | ϕ∨ϕ | ¬ϕ | aϕ | µ(X : τ).ϕ | λ(x : τ).ϕ | ϕ ϕ plus duals ϕ ∧ ψ, [a]ϕ, ν(X : τ) natively But expressions like aP bP not well defined well-formedness condition given by type system

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Types

simple types given via τ ::= Pr complete lattice over transition system T = (S, − →, L) [ [Pr] ] := (2S, ⊆)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Types

simple types given via τ ::= Pr | τ → τ because of right-associativity: τ = τ1 → . . . → τm → Pr each type induces a complete lattice over transition system T = (S, − →, L) using pointwise orderings ⊑ [ [Pr] ] := (2S, ⊆) [ [σ → τ] ] := ([ [σ] ] → [ [τ] ], ⊑)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

Unfolding via σX.ψ = ψ[σX.ψ/X] yields

• λx. x ∨
• µX.λx′. x′ ∨ (X [a]x′)
• [a]x
• P.

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

Unfolding via σX.ψ = ψ[σX.ψ/X] yields

• λx. x ∨
• µX.λx′. x′ ∨ (X [a]x′)
• [a]x
• P.

Using β-reduction we get P ∨

• µX.λx′. x′ ∨ (X [a]x′)
• [a]P.

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

Unfolding via σX.ψ = ψ[σX.ψ/X] yields

• λx. x ∨
• µX.λx′. x′ ∨ (X [a]x′)
• [a]x
• P.

Using β-reduction we get P ∨

• µX.λx′. x′ ∨ (X [a]x′)
• [a]P.

More unfolding: P ∨ (λx′. x′ ∨

• µX.λx′′. x′′ ∨ (X [a]x′′)
• [a]x′) [a]P.

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

Unfolding via σX.ψ = ψ[σX.ψ/X] yields

• λx. x ∨
• µX.λx′. x′ ∨ (X [a]x′)
• [a]x
• P.

Using β-reduction we get P ∨

• µX.λx′. x′ ∨ (X [a]x′)
• [a]P.

More unfolding: P ∨ (λx′. x′ ∨

• µX.λx′′. x′′ ∨ (X [a]x′′)
• [a]x′) [a]P.

More β-reduction: P ∨ [a]P ∨

• µX.λx′′. x′′ ∨ (X [a]x′′)
• [a][a]P) .

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Semantics by Example

Consider (µX.λx. x ∨ (X [a]x)

• P.

Unfolding via σX.ψ = ψ[σX.ψ/X] yields

• λx. x ∨
• µX.λx′. x′ ∨ (X [a]x′)
• [a]x
• P.

Using β-reduction we get P ∨

• µX.λx′. x′ ∨ (X [a]x′)
• [a]P.

More unfolding: P ∨ (λx′. x′ ∨

• µX.λx′′. x′′ ∨ (X [a]x′′)
• [a]x′) [a]P.

More β-reduction: P ∨ [a]P ∨

• µX.λx′′. x′′ ∨ (X [a]x′′)
• [a][a]P) .

We get: P ∨ [a]P ∨ [a][a]P ∨ . . . = n

i=0[a]iP

uniform inevitability!

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Tail-Recursion

Limit use of fp vars: No free fixpoint variables in all subformulas of certain form

• Form ϕ ∧ ψ: no free fixpoint variables in ϕ
• Form [a]ϕ: no free fixpoint variables in ϕ
• Form ¬ϕ: no free fixpoint variables in ϕ
• Form ψ ϕ: no free fixpoint variables in ϕ

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Tail-Recursion

Limit use of fp vars: No free fixpoint variables in all subformulas of certain form

• Form ϕ ∧ ψ: no free fixpoint variables in ϕ
• Form [a]ϕ: no free fixpoint variables in ϕ
• Form ¬ϕ: no free fixpoint variables in ϕ
• Form ψ ϕ: no free fixpoint variables in ϕ

Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Tail-Recursion

Limit use of fp vars: No free fixpoint variables in all subformulas of certain form

• Form ϕ ∧ ψ: no free fixpoint variables in ϕ
• Form [a]ϕ: no free fixpoint variables in ϕ
• Form ¬ϕ: no free fixpoint variables in ϕ
• Form ψ ϕ: no free fixpoint variables in ϕ

Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position µX.p ∨ (a(X ∧ q)) not tail-recursive

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Tail-Recursion

Limit use of fp vars: No free fixpoint variables in all subformulas of certain form

• Form ϕ ∧ ψ: no free fixpoint variables in ϕ
• Form [a]ϕ: no free fixpoint variables in ϕ
• Form ¬ϕ: no free fixpoint variables in ϕ
• Form ψ ϕ: no free fixpoint variables in ϕ

Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position µX.p ∨ (a(X ∧ q)) not tail-recursive µF.λx.[·]⊥ ∨ (F [·]x))⊥ tail recursive

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Tail-Recursion

Limit use of fp vars: No free fixpoint variables in all subformulas of certain form

• Form ϕ ∧ ψ: no free fixpoint variables in ϕ
• Form [a]ϕ: no free fixpoint variables in ϕ
• Form ¬ϕ: no free fixpoint variables in ϕ
• Form ψ ϕ: no free fixpoint variables in ϕ

Result: free fixpoint variables not under any of these operators, each fixpoint definition “morally” contains only one variable (modulo nondeterminism), in operator position µX.p ∨ (a(X ∧ q)) not tail-recursive µF.λx.[·]⊥ ∨ (F [·]x))⊥ tail recursive Thm.: The order (k + 1) model-checking problem for tail-recursive HFL is k-EXPSPACE-complete.

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X]

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X] Over given LTS, µ(: τ)X. ϕ ≡ X m for some m = ht(τ) that is k + 1-fold exponential in size of LTS

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X] Over given LTS, µ(: τ)X. ϕ ≡ X m for some m = ht(τ) that is k + 1-fold exponential in size of LTS → can be stored with k-fold exponentially many bits same for greatest fixpoints

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X] Over given LTS, µ(: τ)X. ϕ ≡ X m for some m = ht(τ) that is k + 1-fold exponential in size of LTS → can be stored with k-fold exponentially many bits same for greatest fixpoints Approach: use top-down local model-checking

• cover fixpoints by unfolding

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X] Over given LTS, µ(: τ)X. ϕ ≡ X m for some m = ht(τ) that is k + 1-fold exponential in size of LTS → can be stored with k-fold exponentially many bits same for greatest fixpoints Approach: use top-down local model-checking

• cover fixpoints by unfolding
• cover ∨, a by free nondeterminism

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound - Preparations

Want algorithm in k-fold exponential space → get nondeterminism for free (Savitch’s Theorem) Consider µ(X : τ). ϕ: Define X 0 = ⊥τ; X i+1 = ϕ[X i/X] Over given LTS, µ(: τ)X. ϕ ≡ X m for some m = ht(τ) that is k + 1-fold exponential in size of LTS → can be stored with k-fold exponentially many bits same for greatest fixpoints Approach: use top-down local model-checking

• cover fixpoints by unfolding
• cover ∨, a by free nondeterminism
• cover ∧, [a], ¬, application via tail-recursion

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ
• an environment η for free variables in ϕ

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ
• an environment η for free variables in ϕ
• a counter cnt specifying degree of unfolding

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ
• an environment η for free variables in ϕ
• a counter cnt specifying degree of unfolding

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ is atomic, return true if s |

= ϕ, false else

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ
• an environment η for free variables in ϕ
• a counter cnt specifying degree of unfolding

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ is atomic, return true if s |

= ϕ, false else

• If ϕ = aϕ′, guess t with s

a

− → t, return check(t, ϕ′, (f1, . . . , fn), η, cnt).

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound

Model-checking whether state t in some LTS is model of ψ procedure check(s, ϕ, (f1, . . . , fn), η, cnt) consumes

• a state s
• a subformula ϕ of ψ
• a list (f1, . . . , fn) of arguments to ϕ
• an environment η for free variables in ϕ
• a counter cnt specifying degree of unfolding

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ is atomic, return true if s |

= ϕ, false else

• If ϕ = aϕ′, guess t with s

a

− → t, return check(t, ϕ′, (f1, . . . , fn), η, cnt).

• . . .
• If ϕ = ϕ1 ∧ ϕ2, → ϕ1 has no free fp vars. Return false if

check(s, ϕ1, (f1, . . . , fn), η, cnt) returns false, return check(s, ϕ2, (f1, . . . , fn), η, cnt) else

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound (cont.)

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ = ¬ϕ′, → ϕ′ has no free fp vars. Return opposite of

check(s, ϕ′, (f1, . . . , fn), η, cnt)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound (cont.)

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ = ¬ϕ′, → ϕ′ has no free fp vars. Return opposite of

check(s, ϕ′, (f1, . . . , fn), η, cnt)

• If ϕ = ϕ′ ϕ′′, → ϕ′′ has no free fp vars. Compute fn+1 =

buildFT(ϕ′′, η), return check(s, ϕ′, (f1, . . . , fn, fn+1), η, cnt)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Upper Bound (cont.)

Behavior of check(s, ϕ, (f1, . . . , fn), η, cnt) depends on form of ϕ:

• If ϕ = ¬ϕ′, → ϕ′ has no free fp vars. Return opposite of

check(s, ϕ′, (f1, . . . , fn), η, cnt)

• If ϕ = ϕ′ ϕ′′, → ϕ′′ has no free fp vars. Compute fn+1 =

buildFT(ϕ′′, η), return check(s, ϕ′, (f1, . . . , fn, fn+1), η, cnt)

• . . .
• If ϕ = µ(X : τ). ϕ′, return

check(s, ϕ′, (f1, . . . , fn), η, cnt[X → ht(τ)]).

• If ϕ = X, X least fp var, return false if cnt(X) = 0, else

return check(s, fp(X), (f1, . . . , fn), η, cnt[X- - ]. buildFT(ϕ, η) calls check(. . . ) again, iterating over all suitable arguments

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors), and initial tiles tI = and T = , and final tile tF =

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors), and initial tiles tI = and T = , and final tile tF = Start with initial 2n

k wide row,

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors), and initial tiles tI = and T = , and final tile tF = Start with initial 2n

k wide row, find matching rows,

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors), and initial tiles tI = and T = , and final tile tF = Start with initial 2n

k wide row, find matching rows,

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Order-k Corridor Tiling Problem

Instance K is set T = { , , . . . , } of tiles, horizontal and vertical matching relations H and V (indicated by colors), and initial tiles tI = and T = , and final tile tF = Start with initial 2n

k wide row, find matching rows, until a row ends

with final tile . . . . . . For k ≥ 0, the order-k corridor tiling problem is k-EXPSPACE hard (van Emde Boas ’97)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Jones’ Encoding of Large Numbers

Observation: Numbers up to 2n

k+1 have at most 2n k bits in binary

Fix an LTS with states S = {0, . . . , n − 1}

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Jones’ Encoding of Large Numbers

Observation: Numbers up to 2n

k+1 have at most 2n k bits in binary

Fix an LTS with states S = {0, . . . , n − 1} Idea (Jones 01): Use states as bits and encode numbers as higher-order functions: 1 2 = 20 + 22 = 5; 1 2 = 21 + 22 = 6

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Jones’ Encoding of Large Numbers

Observation: Numbers up to 2n

k+1 have at most 2n k bits in binary

Fix an LTS with states S = {0, . . . , n − 1} Idea (Jones 01): Use states as bits and encode numbers as higher-order functions: 1 2 = 20 + 22 = 5; 1 2 = 21 + 22 = 6 f with f ( 0 1 2 ) = f ( 0 1 2 ) = 0 1 2 , f ( 0 1 2 ) = f ( 0 1 2 ) = 0 1 2 , f ( 0 1 2 ) = f ( 0 1 2 ) = 0 1 2 , f ( 0 1 2 ) = f ( 0 1 2 ) = 0 1 2 = 25 + 26 = 32 + 64 = 96

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Working with Jones Encodings

Add transitions u 1 2 u u u

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Working with Jones Encodings

Add transitions u, d and e = S × S. 1 2 u u u d d d

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Working with Jones Encodings

Add transitions u, d and e = S × S. 1 2 u u u d d d → Compare numbers: m1 bigger than m2 if ex. a bit set in m1 not set in m2, and all higher bits are set in m1 if they are set in m2: gt0 = λ(m1, m2 : τ0). e

• m2 ∧ ¬m1 ∧ [u](m1 ⇒ m2)
• 12 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Working with Jones Encodings

Add transitions u, d and e = S × S. 1 2 u u u d d d → Compare numbers: m1 bigger than m2 if ex. a bit set in m1 not set in m2, and all higher bits are set in m1 if they are set in m2: gt0 = λ(m1, m2 : τ0). e

• m2 ∧ ¬m1 ∧ [u](m1 ⇒ m2)
• → Increment a number: Set a bit iff it was set and a lower bit was

not set, or if it was not set and all lower bits were set: inc0 = λ(m: Pr). if m then (d¬m) else ([d]m)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Working with Jones Encodings

Add transitions u, d and e = S × S. 1 2 u u u d d d → Compare numbers: m1 bigger than m2 if ex. a bit set in m1 not set in m2, and all higher bits are set in m1 if they are set in m2: gt0 = λ(m1, m2 : τ0). e

• m2 ∧ ¬m1 ∧ [u](m1 ⇒ m2)
• → Increment a number: Set a bit iff it was set and a lower bit was

not set, or if it was not set and all lower bits were set: inc0 = λ(m: Pr). if m then (d¬m) else ([d]m) → Find a number matching a predicate: findk = λ(p : τk+1).

• µ(F : τk → Pr) . λ(m: τk) . ([e](p m)) ∨ F (inck m)
• zerok
• 12 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Encoding the Corridor Tiling Problem

Fix corridor tiling problem K = (T, H, V , tI, t, tF) and extend LTS from previous slides with propositions pI = {0}, p = {|T| − 2}, pF = {|T| − 1} and transitions h = {(i, j): (ti, tj) ∈ H}, v = {(i, j): (ti, tj) ∈ V }. States double as digits and tiles!

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Encoding the Corridor Tiling Problem

Fix corridor tiling problem K = (T, H, V , tI, t, tF) and extend LTS from previous slides with propositions pI = {0}, p = {|T| − 2}, pF = {|T| − 1} and transitions h = {(i, j): (ti, tj) ∈ H}, v = {(i, j): (ti, tj) ∈ V }. States double as digits and tiles!

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Encoding the Corridor Tiling Problem

Fix corridor tiling problem K = (T, H, V , tI, t, tF) and extend LTS from previous slides with propositions pI = {0}, p = {|T| − 2}, pF = {|T| − 1} and transitions h = {(i, j): (ti, tj) ∈ H}, v = {(i, j): (ti, tj) ∈ V }. States double as digits and tiles! Can encode a row of tiles of width 2n

k as function of order 2n k+1

that sends a column number to the singleton set of its tile

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Encoding the Corridor Tiling Problem

Fix corridor tiling problem K = (T, H, V , tI, t, tF) and extend LTS from previous slides with propositions pI = {0}, p = {|T| − 2}, pF = {|T| − 1} and transitions h = {(i, j): (ti, tj) ∈ H}, v = {(i, j): (ti, tj) ∈ V }. States double as digits and tiles! Can encode a row of tiles of width 2n

k as function of order 2n k+1

that sends a column number to the singleton set of its tile Check for final row: isFinal = λ(r : τk). [e]

• (r zerok−1) ⇒ pF
• 13 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Lower Bound

Formula that encodes problem:

• µ(P : τk+1). λ(r1 : τk).(isFinal r1) ∨ (exists succ r1 P)
• init

where exists succ = λ(r1 : τk, p : τk+1). find

• λ(r2 : τk). (horiz r2) ∧ (vert r1 r2) ∧ (p r2).

and horizk verifies horizontal matching in row, and verizk verifies that two rows match

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Lower Bound

Formula that encodes problem:

• µ(P : τk+1). λ(r1 : τk).(isFinal r1) ∨ (exists succ r1 P)
• init

where exists succ = λ(r1 : τk, p : τk+1). find

• λ(r2 : τk). (horiz r2) ∧ (vert r1 r2) ∧ (p r2).

and horizk verifies horizontal matching in row, and verizk verifies that two rows match Not tail-recursive

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

The Lower Bound

Formula that encodes problem:

• µ(P : τk+1). λ(r1 : τk).(isFinal r1) ∨ (exists succ r1 P)
• init

where exists succ = λ(r1 : τk, p : τk+1). find

• λ(r2 : τk). (horiz r2) ∧ (vert r1 r2) ∧ (p r2).

and horizk verifies horizontal matching in row, and verizk verifies that two rows match Not tail-recursive, but can β-reduce:

• µ(P : τk+1). λ(r1 : τk).

(. . . ) ∨

• µ(F : τk+1).λ(r2 : τk)

((. . . ) ∧ (P r2)) ∨ (F (inc r2))

• r1
• init

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Summary

Order-k + 1 tail-recursive HFL captures k-EXPSPACE (as opposed to k + 1-EXPTIME for full k + 1-HFL)

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Summary

Order-k + 1 tail-recursive HFL captures k-EXPSPACE (as opposed to k + 1-EXPTIME for full k + 1-HFL) Remark: Instead of unlimited ∨, , can also use unlimited ∧, [], but restricted ∨,

15 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Summary

Order-k + 1 tail-recursive HFL captures k-EXPSPACE (as opposed to k + 1-EXPTIME for full k + 1-HFL) Remark: Instead of unlimited ∨, , can also use unlimited ∧, [], but restricted ∨, ; can even mix both if “interface” right, i.e., free fp vars separated

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Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Summary

Order-k + 1 tail-recursive HFL captures k-EXPSPACE (as opposed to k + 1-EXPTIME for full k + 1-HFL) Remark: Instead of unlimited ∨, , can also use unlimited ∧, [], but restricted ∨, ; can even mix both if “interface” right, i.e., free fp vars separated Further research: Look whether/how much tail-recursion requirement can be removed from low-type fixpoints (probably not entirely)

15 / 15

Bruse, Lange, Lozes: Space-Efficient Fragments of Higher-Order Fixpoint Logic

Summary

Order-k + 1 tail-recursive HFL captures k-EXPSPACE (as opposed to k + 1-EXPTIME for full k + 1-HFL) Remark: Instead of unlimited ∨, , can also use unlimited ∧, [], but restricted ∨, ; can even mix both if “interface” right, i.e., free fp vars separated Further research: Look whether/how much tail-recursion requirement can be removed from low-type fixpoints (probably not entirely) Thanks!

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